The reaction K−P → ΛK+K− and the generalized Veneziano model

Nuclear Physics B28 (1971) 85-96. North-Holland Publishing Company

THE R E A C T I O N K - P --o A K + K A N D THE GENERALIZED V E N E Z I A N O MODEL P. A. S C H R E I N E R :~ and D. H. S T O R K

Department of Physics, University of California, Los Angeles, California, USA R. T. ROSS, A. G. C L A R K and L. LYONS

Department of Nuclear Physics, University of Oxford, Oxford, England Received 30 November 1970

Abstract: New data for the reaction K-p-* AK+K- at 3.3 GeV/c have been used to investigate the five-particle generalized Veneziano model. The amplitude is constrained by symmetry properties of the crossed channels and by non-exchange degeneracy of the N~ trajectory. The H a r a r i - R o s n e r dual quark model determines the trajectories which couple to the K+K- system. Reasonable choices for the other t r a j e c t o r i e s lead to good agreement with these data as well as data at higher and lower beam momentum. The pomeron exchange contribution to the reaction is shown to be small; however, both pseudoscalar and vector kinematic factors are required for detailed agreement with the angular distributions. {- The •

.

.

~-

+

m o d e l fails to account for the c r o s s m g - s y m m e t r l creaction K p-~ AK K .

I. INTRODUCTION

We have i n v e s t i g a t e d the r e a c t i o n K - p ~ AK+K -

(1)

by m e a n s of the f i v e - p a r t i c l e V e n e z i a n o m o d e l [1]. T h e a m p l i t u d e f o r t h i s r e a c t i o n i s c o n s t r a i n e d by the s y m m e t r y p r o p e r t i e s of c r o s s e d c h a n n e l s s u c h as K+p ~ AK+K+

(2)

:#

and by n o n - e x c h a n g e d e g e n e r a c y of the N a t r a j e c t o r y . We show that the H a r a r i - R o s n e r d u a l q u a r k m o d e l [2] m a k e s c e r t a i n p r e d i c t i o n s f o r the K+K c o u p l i n g w h i c h a r e w e l l s a t i s f i e d when the V e n e z i a n o m o d e l i s c o m p a r e d with experiment. D a t a f o r r e a c t i o n (1) a r e s p a r s e . We c o n s i d e r p u b l i s h e d d a t a at b e a m mom e n t u m b e t w e e n 2 and 6 G e V / c (ref. [3]), b u t c o n c e n t r a t e on a p r e l i m i n a r y Work supported in part by the US Atomic Energy Commission.

86

:P.A.SCHREINEI~ et al. K-p--AK+K -

5 5 GeV/c

o

100 5O 0.9 °J(D

>~ L9

o

(b)

I.I 1.5 moss (K+K -) GeV/c-2 '

1.5

'

30

Q

20 ~> I0

1.6 (c)

1.8 2.0 moss ( A K +) GeV/c2 I

I

2.2 I

10 1.6

1.8 2.0 moss ( A K - ) GeV/cz

2.2

Fig. 1. Histograms of the K+K -, AK + and AK- effective m a s s e s for 322 weighted events of reaction (1) at 3.3 GeV/c. Cross hatched a r e a s r e p r e s e n t ~b(1020) events: M(KK) --< 1.04 GeV. The curves r e p r e s e n t fit 3 to eq. (6). s a m p l e of 287 e v e n t s a t a b e a m m o m e n t u m of 3.30 G e V / c $. T h e l a t t e r d a t a a r e p a r t of an O x f o r d - U C L A e x p e r i m e n t p e r f o r m e d w i t h t h e C E R N t w o - m e t e r l i q u i d - h y d r o g e n b u b b l e c h a m b e r . F i g . 1 s h o w s t h e K+K - , K+A and K - A , e f f e c t i v e - m a s s d i s t r i b u t i o n s f o r t h e s e d a t a . S t r o n g ~(1020) p r o d u c t i o n i s o b s e r v e d in t h e K+K - s p e c t r u m ; t h e s e ~ - e v e n t s [M(K÷K -) < 1.040 GeV] a r e c r o s s - h a t c h e d in fig. l b and l c . A t the p r e s e n t l e v e l of s t a t i s t i c s , no s i g n i f i c a n t N* o r ~ * p r o d u c t i o n can b e o b s e r v e d in t h e r e s p e c t i v e d i s t r i b u t i o n s . E v e n a t 5.5 GeV/c, N* p r o d u c t i o n i s n e g l i g i b l e i n d i c a t i n g t h a t t h e p o m e r o n e x c h a n g e c o n t r i b u t i o n to r e a c t i o n (1) i s not l a r g e ( s e e s e c t . 4).

~t All of these events have a visible A --*p ~ - decay in the chamber. The data shown in figs. 1, 3 and 4 have been weighted to account for short (~ 5 mm) and long (outside fiducial volume) decays; the average weight is 1.12.

THE REACTION K-p -* AK+K-

(a)

_

_~P

~,P~4_

87

(c)

Fig. 2. The four particle graphs that do not require exotic trajectories (all lines are drawn outgoing) and their corresponding Harari-Rosner dual quark diagrams. 2. MODEL The model used for our s c a t t e r i n g amplitude is

M = N K ( ~ G i B5i) ,

(3)

w h e r e N and G i a r e real n u m b e r s , K is a kinematic f a c t o r , the summation is o v e r the twelve n o n - c y c l i c p e r m u t a t i o n s of the external p a r t i c l e s and B 5 is the B a r d a k c i - R u e g g - V i r a s o r o amplitude [1]. F o r r e a c t i o n (1) as well as the c r o s s e d reaction (2), eight of the twelve p e r m u t a t i o n s contain exotic Regge t r a j e c t o r i e s and will be neglected. The r e m a i n i n g p e r m u t a t i o n s a r e shown as graphs A to D in fig. 2. The c o r r e sponding "dual q u a r k s t r u c t u r e s " a r e also shown; all four quark s t r u c t u r e s a r e p l a n a r , and may be expected to contribute to the amplitude [2]. 2.1. S y m m e t r y r e q u i r e m e n t s An i m p o r t a n t c o n s t r a i n t in our model a r i s e s f r o m the r e q u i r e m e n t of s y m m e t r y with r e s p e c t to the i n t e r c h a n g e of two identical bosons in the c r o s s e d channels. F o r the r e a c t i o n K+p -~ AK+K+, the final state K+K+ s y s tem is in an even isospin state and, t h e r e f o r e , the spacial p r o p e r t i e s of o u r amplitude M m u s t be even. Graph A d i f f e r s f r o m B only in the i n t e r change of the two identical K+ lines and s i m i l a r l y for graphs C and D. Consequently, if the k i n e m a t i c f a c t o r K is even, GA = GB and GC = GD; if K is odd, GA = -G B and GC = -G D. The line r e v e r s a l p r o p e r t i e s of the B 5 a m p l i tude f u r t h e r n e c e s s i t a t e the application of these s y m m e t r y conditions to r e action (1).

2.2. T r a j e c t o r i e s We now c o n s i d e r the Regge t r a j e c t o r i e s a s s o c i a t e d with the v a r i o u s channels: (i) The existence of a dominant i s o s p i n - z e r o , AO(1020) s u b - c h a n n e l , and the o b s e r v a t i o n by Schmid [4] that the Y~(1385) is not strongly coupled to the Kp s y s t e m , indicate that we should couple the K p s y s t e m to the A(1115)

88

P.A. SCHREINER et al.

K - p ~/kK+K -

5.5 GeV/c

60

E ~20 -I0

cosOA*

I0

-10 COS8K~+

10

-I.0 COSeK~- 10

Fig. 3. Histograms of the c.rn. production angles (relative to the incident K-) of outgoing A, K÷ and K- for reaction (1) at 3.3 GeV/c. Cross hatched events r e p r e sent q5(1020) events. The curves r e p r e s e n t fit 3 to eq. (6). K-p~A~oeo

3.5 GeV/c i

11)

20 ,o i

-I0

0.0 w

I0

180

36O

COS~j

Fig. 4. Histograms of the Jackson and Treiman-Yang decay angles of q5(1020) events at 3.3 GeV/c; the angles a r e those of the outgoing K-. The curves r e p r e sent fit 3 to eq. (6). t r a j e c t o r y . T h e r e s u l t s of t h i s p a p e r a r e q u a l i t a t i v e l y s i m i l a r if a Y~(1385) trajectory is used instead. (ii) We t a k e t h e A K + s y s t e m to b e c o u p l e d to the N a t r a j e c t o r y . (iii) T h e A K - s y s t e m m u s t b e c o u p l e d to a - ~ - t r a j e c t o r y . E x c e p t f o r the - ( 1 3 2 0 ) and E(1530) s t a t e s , no d e f i n i t e s p i n - p a r i t y d e t e r m i n a t i o n s h a v e b e e n m a d e on o b s e r v e d S = -2 m a s s e n h a n c e m e n t s . F o r o u r f i t s we h a v e c h o s e n a - t r a j e c t o r y p a s s i n g t h r o u g h a m a s s of 1. 530 GeV a t a J = ~ and w i t h a s l o p e of 1.0. T h i s t r a j e c t o r y s h o u l d b e c o n s i d e r e d a s r e p r e s e n t a t i v e of a r a n g e of p o s s i b l e t r a j e c t o r i e s [5]. It c l o s e l y c o r r e s p o n d s f o r e x a m p l e to o n e g i v e n by S p e c t o r [6] in a d i s p e r s i o n - t h e o r y a n a l y s i s . (iv) T h e A~ s y s t e m can b e c o u p l e d to b o t h t h e d e g e n e r a t e K ( 4 9 4 ) - K(1300) and K*(892) - K * ( 1 4 2 9 ) t r a j e c t o r i e s . T h e d e n s i t y m a t r i x e l e m e n t s f o r q~(1020) p r o d u c t i o n in r e a c t i o n (1) at m e d i u m and h i g h - b e a m m o m e n t u m s u g g e s t t h a t t h e K* t r a j e c t o r y d o m i n a t e s , but w e w i l l c o n s i d e r both. (v) T h e K+K - s y s t e m can be c o u p l e d to b o t h the p - A 2 and ~) - f ' t r a j e c t o r i e s and we i n v e s t i g a t e b o t h in d e t a i l ( s e e s u b s e c t . 3.1). T h e p a r a m e t e r s u s e d f o r a l l the t r a j e c t o r i e s a r e g i v e n in t a b l e 1. W e f o l l o w P e t e r s s o n et al. [7] in a l t e r i n g the i n t e r c e p t of b a r y o n e x c h a n g e t r a -

89

THE REACTION K-p -~ AK+KTable 1 T r a j e c t o r i e s and B 5 arguments. Particles

Channel

Trajectory

Re (~)

K-p

s

A(1115)

-0.68 +0.95 s -0.68+0.95 t

0.0

1-a

~ (1530)

-0.84+ 1.00 s

0.11 ( s - 2.24)

~-a

-0.84+ 1.00 t

0.0

1-~

-0.37 + 1.00 s

0.14 ( s - 1.0)

½-~ 1-~

t K-A

s t

K+

s

N(940)

t t

K+K-

Im (a)

X

0.14 ( s - 2.1)

½-a

-0.37 + 1.00 t

0.0

K(494)

-0.15+0.68 t

0.0

-~

K(892)

0.34+0.82 t

0.0

1-q

s

~b

0.17 + 0.80 s

0.065(s - 0.977)

1-

s

P

0.48+0.90 s

0.13 ~/s -0.072

1-c~

t

~

0.17+0.80t

0.0

1-~

t

p

0.48 + 0.90 t

0.0

1-

j e c t o r i e s in o r d e r to r e s t o r e c o r r e c t R e g g e b e h a v i o u r i n the c o r r e s p o n d i n g B 5 amplitudes.

2.3. Kinematic factors The k i n e m a t i c f a c t o r K in eq. (3) d e p e n d s upon the c h o i c e of the AD t r a j e c t o r y . F o r the K*(892) - K*(1420) c o u p l i n g we u s e the v e c t o r - e x c h a n g e k i n e m a t i c f a c t o r [7]

T h i s f a c t o r i s a n t i - s y m m e t r i c u n d e r the i n t e r c h a n g e of the two i d e n t i c a l K+ l i n e s , so that the r e q u i r e d s y m m e t r y of o u r a m p l i t u d e f o r c e s GA = - G B and GC = - G D. Note t h a t t h i s c o n d i t i o n r e s u l t s in A~ and A K - c o u p l i n g to n o n e x c h a n g e d e g e n e r a t e K*(1420) and _=(1830) t r a j e c t o r i e s r e s p e c t i v e l y . To o b t a i n d e t a i l e d a g r e e m e n t b e t w e e n the m o d e l and e x p e r i m e n t at s m a l l p r o d u c t i o n a n g l e s (cos a~ < -0.9) and with the ~b(1020) d e c a y a n g l e s , o n e m u s t a l s o i n c l u d e K(494) - K ( 1 3 0 0 ) c o u p l i n g to the A~ s y s t e m . To i n t r o d u c e this t r a j e c t o r y we u s e the k i n e m a t i c f a c t o r d e v e l o p e d in ref. [8] f o r p s e u d o s c a l a r e x change KpS

:

PK PK ,

(51 -

+

w h e r e the t h r e e - m o m e n t u m v e c t o r s P a r e e v a l u a t e d in the K K2 r e s t f r a m e . S i n c e this k i n e m a t i c f a c t o r i s s y m m e t r i c u n d e r the i n t e r c h a n g e of the two K+, we s e t GA = GB and G C = GD i n the p s e u d o s c a l a r c o n t r i b u t i o n to the amplitude. The r e s u l t i n g A~ and AK c o u p l i n g s i n t h i s c a s e a r e to n o n - e x c h a n g e d e g e n e r a t e s K(494) a n d ~ ( 1 5 3 0 ) t r a j e c t o r i e s r e s p e c t i v e l y . If o n e a l s o

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P . A . SCHREINER et al.

assumes that the N a trajectory is non-exchange degenerate, G B = GD. T h e c o m p l e t e a m p l i t u d e s q u a r e d i s t h e r e f o r e

i 12

G A = G C and

2 (6)

where the V, PS subscripts refer to the K*, K trajectories for the A~ system in graphs A and B.

3. F I T T I N G THE D A T A In f i t t i n g t h i s m o d e l to o u r d a t a w e w a n t to e x a m i n e in d e t a i l the t r a j e c t o r y a m b i g u i t y in the K+K - s y s t e m . T h e s y m m e t r y of the f i v e - p o i n t f u n c t i o n s i m p o s e s t h r e e c o n s t r a i n t s on the c h o i c e of t h e s e t r a j e c t o r i e s . T h e s e c o n s t r a i n t s r e l a t e the two K+K - t r a j e c t o r i e s of g r a p h A to t h o s e of g r a p h B and the K+K - t r a j e c t o r y of g r a p h C to t h a t of g r a p h D a s f o l l o w s :

-- a(K K-)D

(7)

If e a c h K+K - s y s t e m c o u p l e s to e i t h e r t h e p - o r ~ - t r a j e c t o r y , t h e r e a r e 23 = 8 p o s s i b l e c h o i c e s f o r the t r a j e c t o r i e s of the f o u r g r a p h s . To r e s o l v e t h i s a m b i g u i t y , a l e a s t - s q u a r e s f i t w a s p e r f o r m e d f o r e a c h of the e i g h t c o m b i n a t i o n s to the f o l l o w i n g e x p e r i m e n t a l d i s t r i b u t i o n s of r e a c t i o n (1) a t 3.3 G e V / c : the K+K-~ A K + and A K - e f f e c t i v e m a s s e s , the A , K + and K - c . m . p r o d u c t i o n a n g l e s , and the J a c k s o n and T r e i m a n - Y a n g d e c a y a n g l e s of t h e (p(1020). T h e p a r a m e t e r s N 2 and ~, w e r e a l l o w e d to v a r y f o r e a c h of t h e e i g h t t r a j e c t o r y c o m b i n a t i o n s . T h e r e s u l t s of the f i t s to the 111 h i s t o g r a m b i n s a r e ~ v e n in t a b l e 2, w h e r e × 2 c o r r e s p o n d s to t h e c h i 2 v . s q u a r e d f o r ~ = 0.0 and × V + PS c o r r e s p o n d s to t h e c h l - s q u a r e d f o r the l i s t e d y. T h e b e s t f i t i s g i v e n by c o m b i n a t i o n 3; f i t s 5 and 7 a r e i n f e r i o r to t h a t of 3 b u t c a n n o t b e d e c i s i v e l y r e j e c t e d . T h e m a i n d e f e c t in a l l of the f i t s e x c e p t 3 o c c u r s in the c . m . p r o d u c t i o n a n g l e s ; in p a r t i c u l a r , l a r g e e x c e s s e s of e v e n t s a r e p r e d i c t e d f o r c o s 0 ~ > 0. W e c o m p a r e f i t 3 of t a b l e 2 w i t h r e a c t i o n (1) in fig. 1 and f i g s . 3-6. F o r o u r 3.3 GeV d a t a , t h e a g r e e m e n t w i t h the e f f e c t i v e - m a s s d i s t r i b u t i o n s i s good $ , and i t r e m a i n s s o a f t e r r e m o v a l of the ~b(1020) e v e n t s a s shown in In our model the width of the ~(1020) is 5 MeV; however, in o r d e r to compare the model with data we have histogrammed our Monte Carlo events (weighted by the Veneziano amplitude) in bin sizes equal to those of the data; it is this procedure which results in the apparent ~ 20 MeV width in fig. l a .

THE REACTION K-p ~ AK+K-

91

Table 2 Combinations of the three trajectories coupled to the K+K- system (al, a2, a3) and the minimum chi-squared for fits to eq. (6). FIT

aI

a2

a3

X~

2 PS XV+

1

p

p

p

2403

2263

0.038

2

~

p

p

663

471

0.060

3

p

~

p

469

233

0.059

4

p

p

~

1310

1289

0.018

5

~

~

p

691

365

0.110

6

~

p

¢

928

690

0.077

7

P

~

~

653

292

0.082

8

¢

~

¢

soo

413

0.130

2 2 The XV is the result for ~ = 0.0 and the XV + PS is the result for the fitted value of )' in the table. The third fit corresponds to the Harari-Rosner dual quark model. fig. I b and Ic. Fig. 3 d i s p l a y s the c.m. p r o d u c t i o n a n g u l a r d i s t r i b u t i o n s f o r the A, K+ and K- at 3.3 G e V / c . As in fig. I , the c r o s s - h a t c h e d a r e a s (and the l o w e r c u r v e s ) r e p r e s e n t the G - e v e n t s ; no d i s c r e p a n c i e s a r e a p p a r e n t . The d e c a y a n g u l a r d i s t r i b u t i o n s of the ~b a r e shown in fig. 4a and 4b. The J a c k s o n angle i s w e l l d e s c r i b e d b e c a u s e of the p r e s e n c e of both Ap t r a j e c t o r i e s ; the T r e i m a n - Y a n g d i s t r i b u t i o n i s only q u a l i t a t i v e l y r e p r o d u c e d . 3. I. Q u a r k s t r u c t u r e The b e s t fit h a s the t r a j e c t o r y c o m b i n a t i o n

a1

ap,

=

~2 = a ¢ , a 3 = Up 9

and c o r r e s p o n d s to a p r e d i c t i o n of the H a r a r i - R o s n e r dual q u a r k model. T h i s r e s u l t i s e a s i l y s e e n by i n s p e c t i n g fig. 2 and r e m e m b e r i n g that with a v e c t o r m e s o n m i x i n g a n g l e of tan - I (1/4~), the q u a r k contents of the q~ and p a r e [9] ~ -X~,,

P ~2

(p~ - n ~ ) .

The a g r e e m e n t of the d a t a with the q u a r k - m o d e l p r e d i c t i o n f o r the t r a j e c t o r i e s of the B 5 f u n c t i o n s can be u n d e r s t o o d i n t u i t i v e l y as follows. A m a i n f e a t u r e of the A ~ final s t a t e i s that t h e r e is a l m o s t no b a r y o n e x c h a n g e , i . e . v e r y few e v e n t s p r o d u c e d in the b a c k w a r d d i r e c t i o n . But if the K-K~ s y s t e m in g r a p h D i s coupled to the ~ , then t h e r e would be a s i z e a b l e b a r y o n -

P. A. SCHREINER et al.

92

e x c h a n g e c o n t r i b u t i o n f r o m D - to q S - p r o d u c t i o n . T h u s a ( K - K ~ ) D -= a3 m u s t b e c o u p l e d to the p. A s i m i l a r a r g u m e n t c o n c e r n i n g b a r y o n e x c h a n g e in ~ p r o d u c t i o n f r o m g r a p h B l e a d s to a ( K - K ~ ) B --- a l = ap" F i n a l l y , s i n c e qSp r o d u c t i o n p r o c e d i n g by m e s o n e x c h a n g e i s a p r o m i n e n t f e a t u r e of o u r d a t a , the o n l y r e m a i n i n g K+K - t r a j e c t o r y a ( K - K ~ ) A --- a 2 m u s t b e c o u p l e d to the qS. T h e a g r e e m e n t of t h i s q u a r k p r e d i c t i o n c o n t r a s t s w i t h a p r e v i o u s B 5 s t u d y of K - p ~ A~+~ - [7] w h i c h found t h a t the H a r a r i - R o s n e r m o d e l i s not successful. 3.2. O t h e r a s p e c t s To d e m o n s t r a t e how the m o d e l b e h a v e s a s a f u n c t i o n of i n c i d e n t b e a m m o m e n t u m , we d i s p l a y in fig. 5 the K+K - e f f e c t i v e m a s s and q~ c . m . p r o d u c tion a n g u l a r d i s t r i b u t i o n s a t 2.24 and 5.5 G e V / c ; the a g r e e m e n t i s good. F i g . 6 s h o w s the v a r i a t i o n of the t o t a l c r o s s s e c t i o n w i t h b e a m m o m e n t u m ; e x c e p t f o r b e i n g s o m e w h a t l a r g e a t ~ 6 G e V / c , the p r e d i c t i o n i s in r e a s o n a b l e a g r e e m e n t w i t h e x p e r i m e n t . T h e s m a l l o s c i l l a t i o n at a b o u t 2.3 G e V / c i s f r o m the p s e u d o s c a l a r c o n t r i b u t i o n to the , amplitude. F i n a l l y , we w i s h to e m p h a s i z e t h a t if N a n o n - e x c h a n g e d e g e n e r a c y and the s y m m e t r y r e q u i r e m e n t s of the a m p l i t u d e a r e r e m o v e d , then s a t i s f a c t o r y f i t s can be o b t a i n e d f o r s o m e of the o t h e r K+K - t r a j e c t o r y c o m b i n a t i o n s . In p a r t i c u l a r if o n l y the ~ - t r a j e c t o r y i s u s e d , a f i t v e r y s i m i l a r to the o n e shown in the f i g u r e s i s o b t a i n e d when GA = 2 G c , G B = GD = O, and ~ = 0.05.

4. THE R E A C T I O N K+p -* AK+K + A s a c h e c k of c r o s s i n g s y m m e t r y we n e x t c o n s i d e r r e a c t i o n (2). H o w -

4ol p~~(~2o) 5.5GeV/c20~ ,

P~b= 5.5 GeV/c

,°I

2O

2.3

-I.0

4.3

lO

20~j P~b= 2.24 I0 ]1

O.N

,

I

,

~ - L n

rn

I.~ 1.362 ~ mass squared (K+K-) ~ / c )':

-I.0

1.0 COSOA ~

Fig. 5. Histograms of K+K - effective m a s s for reaction (1) at a beam momentum of 5.5 GeV/c, ~b(1020) c.m. production angle at 5.5 GeV/c, K+K- effective m a s s for reaction (1) at a beam momentum of 2.24 GeV/c, and ~b(1020) c.m. production angle at 2.24 GeV/c. The curves r e p r e s e n t fit 3 to eq. (6).

THE

REACTION

K-p

-~ AK+K

-

93

'. K - p ' _AK÷t~ x Kip ~AK~'K *

80

t

~ 4O b

2O

4

~

8

momentum (GeV/c) Fig. 6. Cross sections of reaction (1) and (2) as a function of beam momentum. Upper solid curve r e p r e s e n t s fit 3 to eq. (6); lower solid curve and its n o r m a l i zation is obtained from c r o s s i n g reaction (1); dashed curve r e p r e s e n t s the upper l i m i t to the pomeron exchange contribution to the reactions. e v e r , b e c a u s e K p s a s g i v e n by eq. (5) i s not c r o s s i n g s y m m e t r i c , o n l y the . f i r s t ( v e c t o r ) p a r t of t h e a m p l i t u d e g i v e n by eq. (6) can b e c r o s s e d . W e h a v e p e r f o r m e d t h i s c r o s s i n g p r o c e d u r e f o r r e a c t i o n (2) and s h o w the c o n t r i b u tion to the t o t a l c r o s s s e c t i o n in fig. 6, t o g e t h e r w i t h the m e a s u r e d v a l u e s [10]. O n l y a s m a l l p a r t of the K+p c r o s s s e c t i o n h a s b e e n a c c o u n t e d f o r in t h i s way. A t high m o m e n t u m , t h e e x p e r i m e n t a l m a s s s p e c t r u m and p r o d u c tion a n g l e s a g r e e q u a l i t a t i v e l y w i t h t h e m o d e l ; h o w e v e r , a t low m o m e n t u m ~ 3 GeV/c there is poor agreement with the production angles. It i s of i n t e r e s t to i n v e s t i g a t e w h e t h e r t h i s f a i l u r e f o r t h e AK+K + f i n a l s t a t e i s d u e to f e a t u r e s w e h a v e e x p l i c i t l y n e g l e c t e d ( p o m e r o n and p s e u d o s c a l a r e x c h a n g e s ) o r to m o r e i n h e r e n t d e f i c i e n c i e s of t h e a m p l i t u d e (e.g. t h e p r e s e n c e of f e r m i o n and b a r y o n t r a j e c t o r i e s ) . W e h a v e t h e r e f o r e a t t e m p t e d to e s t i m a t e t h e i m p o r t a n c e of p o m e r o n and p s e u d o s c a l a r e x c h a n g e s in r e a c t i o n (2).

4.1. Pomeron exchange F o r the p o m e r o n - e x c h a n g e c o n t r i b u t i o n , the m o d e l of P o k o r s k i and S a t z [11] w a s c o n s i d e r e d . F i g . 7 s h o w s the t h r e e a l l o w e d p o m e r o n e x c h a n g e g r a p h s ; f o r the /th g r a p h , t h e a m p l i t u d e i s

U P =glS[(UA - M p ) 2 - t A p ] ½ e b / K K B4(~1 - al, ~2 - a2),

(8)

whereg/is t h e c o u p l i n g c o n s t a n t f o r g r a p h l, s i s t h e t o t a l c . m . e n e r g y s q u a r e d , a I and a 2 a r e the R e g g e t r a j e c t o r i e s w h i c h c o u p l e to t h e p a i r s of p a r t i c l e s a t t h e l o w e r v e r t e x , and ~i i s the l e a d i n g p o l e on a/. A s in t h e B 5 m o d e l d e s c r i b e d a b o v e , the A K t r a j e c t o r y c o u p h n g i s t a k e n to t h e NED N a. T h i s r e q u i r e s t h a t g A = g B . F o r t h e K+p s y s t e m we u s e the A ( l l 1 5 ) t r a j e c t o r y . W e t a k e t h e K(494) t r a j e c t o r y f o r the A p e o u p i i n g s i n c e i t c o u p l e s to t h e p o m e r o n - K + s y s t e m w h i l e the K*(890) d o e s not. -~-

.

.

.

94

P.A. SCHREINER et al. K+ -

K+ -

p ~

~""/k

K +-

K+ -

K-

p

K+

A "'-

A

B

K-+

~"p C

Fig. 7. Schematic diagrams of the reactions K±p ~ AK+K± proceeding by means of pomeron exchange. With r e g a r d to the relative weight of graph C c o m p a r e d to that of A and B, two limiting c a s e s have been investigated: (i) (ii)

gA = gB = g

and

g c = 0,

gA : gB : g c : g"

They c o r r e s p o n d r e s p e c t i v e l y to ED and NED coupling of the A(ll15) and K(494) t r a j e c t o r i e s . The l a t t e r possibility is consistent with the quark s e lection r u l e s of C a r l i t s et al. [12] which forbid p o m e r o n - k a o n coupling to K(1300). In each case, the amplitudes given by eq. (8) w e r e added, squared, and then added incoherently to that of eq. (6). Fits were again p e r f o r m e d to our 3.3 G e V / c data. The p a r a m e t e r b was taken to be 2.0 (ref. [13]', the K+K - couplings in the B 5 functions w e r e taken f r o m the quark model, and N2, ), and g2 w e r e free p a r a m e t e r s . In each case the r e s u l t s a r e quite s i m i l a r to the fit 3 shown in figs. 1, 3 and 4. The inclusion of p o m e r o n exchange in the amplitude does not lead to significant i m p r o v e m e n t in the fits [Xz = 196 for case (i) and ×2 = 209 for case (ii)]. Thus, the fitted p o m e r o n contributions (16% and 12% respectively) a r e r a t h e r i m p r e c i s e l y determined. In fact, when extrapolated to 12.7 GeV/c; they a r e 34 #b, and 23 #b, which exceed the experimental total c r o s s s e c tion [14] of 12 + 3 #b. By r e n o r m a l i z i n g the p o m e r o n contribution to this exp e r i m e n t a l value we obtain an upper limit of g r e a t e r significance than the fitted r e s u l t s d e s c r i b e d above. This upper limit to the p o m e r o n c r o s s s e c tion, which should apply to both AK+K - and AK+K+, is shown as the dashed curve in fig. 6. 4.2. P s e u d o s c a l a r e x c h a n g e The p s e u d o s c a l a r K-exchange contributions to r e a c t i o n s (1) and (2) a r e s c h e m a t i c a l l y illustrated in fig. 8. The c o r r e s p o n d i n g s c a t t e r i n g amplitude is approximately given by 1 tAp _ ~

B4(1 - ap, 1 - adp ) .

In the K+p reaction, the ~b a p p e a r s only in the t-channel while for K-p it a p p e a r s in an s-channel. This r e s u l t s in the B 4 contribution to the c r o s s section being o v e r two o r d e r s of magnitude l a r g e r in the K-p than in the

THE REACTION K-p ~ AK+K K-

K-

K+

95 K+

p~ ~A p ~ ~A Fig. 8. Schematic diagrams of the reactions K±p -~ A K + K ± proeeding by m e a n s of K(494) exchange.

K+p r e a c t i o n . S i n c e p s e u d o s c a l a r e x c h a n g e a c c o u n t s f o r o n l y ~ -~ of r e a c t i o n (1) a t 3.3 G e V / c , i t s c o n t r i b u t i o n to r e a c t i o n (2) w o u l d a p p e a r to b e n e g l i gible. W e t h u s t e n t a t i v e l y c o n c l u d e t h a t , if the p o m e r o n and p s e u d o s c a l a r e x • . + 4+ . . c h a n g e c o n t r t b u h o n s to K p -~ A K K a r e e s t t m a t e d m the m a n n e r d e s c r i b e d a b o v e , n e i t h e r of t h e m i s of s u f f i c i e n t i m p o r t a n c e to e x p l a i n the d i f f e r e n c e b e t w e e n the K+p d a t a and the B 5 a m p l i t u d e o b t a i n e d by c r o s s i n g f r o m AK+K -. It i s t h e r e f o r e c o n c e i v a b l e t h a t t h e B 5 f u n c t i o n a n d / o r the v e c t o r e x c h a n g e k i n e m a t i c f a c t o r a r e at fault. C o n c e r n i n g the e f f e c t of b a r y o n s p i n , w e o n l y n o t e t h a t f o r r e a c t i o n (2) t h e r e a r e t h r e e t - c h a n n e l b a r y o n t r a j e c t o r i e s in g r a p h s C and D.

5. CONCLUSIONS (i) T h e V e n e z i a n o m o d e l g i v e s a good d e s c r i p t i o n of K - p ~ AK+K - a t a l l available beam momentum. (ii) W h e n t h e r e l a t i v e c o n t r i b u t i o n s of the t e r m s in the a m p l i t u d e a r e t h o s e r e q u i r e d by t h e s y m m e t r y of c r o s s e d r e a c t i o n s , the R e g g e t r a j e c t o r i e s u s e d in t h e m o d e l a r e c o n s i s t e n t w i t h t h o s e s u g g e s t e d by the H a r a r i Rosner quark model. (iii) T h e AK+K + f i n a l s t a t e i s n o t c o r r e c t l y d e s c r i b e d by o u r v e c t o r B 5 a m p l i t u d e . T h i s d e f i c i e n c y d o e s not a p p e a r to b e e x p l i c a b l e in t e r m s of p s e u d o s c a l a r a n d / o r p o m e r o n e x c h a n g e s , but m a y be d u e to a f a i l u r e of the m o d e l to a c c o u n t f o r b a r y o n e x c h a n g e .

REFERENCES [1] K. Bardak£i and H. Ruegg, Phys. L e t t e r s 28B (1968) 342; M . V i r a s o r o , Phys. L e t t e r s 22 (1969) 37. [2] H. H a r a r i , Phys. Rev. L e t t e r s 22 (1969) 562; J. Rosner, Phys. Rev. L e t t e r s 22 (1969) 689. [3] P. Schlein, W. E. Slater, L . T . Smith, D. H. Stork and H. K. Ticho, Phys. Rev. L e t t e r s 10 (1963) 368; G. W. London, R.R. Ran, N. P. Samios, S.S. Yamamoto, M. Goldberg, S. Lichtman, M. P r i m e r and J. Leitner, BNL 9542 (C-58) 1965; Birmingham-Glasgow-London (I.C.)-Oxford-Rutherford Collaboration, Phys. Rev. 152 (1966) 1148; J. Lindsey and G. Smith, Phys. Rev. 147 (1966) 913; Birmingham-Glasgow-London (I.C.)-Oxford-Rutherford Collaboration, Nuovo Cimento 53A (1968) 522; J. Mott, R. A m m a r , R.Davis, W. Kropac, D.Slate, B. W e r n e r , S. Dagan, M. D e r rick, T. F i e l d s , J. Loken and F. Schweingruber, Phys. Rev. 177 (1968) 1966.

96

P . A . SCHREINER et al.

[4] C. Schmid, Nuovo Cim. L e tt e r s 1 (1969) 165. [5] Aachen-Berlin-CERN-London-Vienna Collaboration, Phys. Letters 28B (1969) 439. [6] R. Spector, Phys. Rev. D1 (1970) 365. [7] B. P e t e r s s o n and N.T~Srnqvist, Nucl. Phys. B13 (1969) 629. [8] P. A. Schreiner, D.H. Stork, A. G. Clark, D. Saxon and L. Lyons, Nucl. Phys. B24 (1970) 157. [9] J. Kokkedee, The quark model (Benjamin, New York, 1969). [10] M. F e r r o - L u z z i , R. George, Y. Goldschmidt-Clermont, V. P. Henri, B. Jongej am s , D.W.G. Leith, G. R. Lynch, F. Muller and J. M. P e r r e a u , Phys. Letters 17 (1965) 155; A. R. Erwin, W.D. Walker and A. Weinberg, Phys. Rev. Letters 16 (1966) 1063; T. J o l d e r s m a , R.B. P a l m e r and N. P. Samios, Phys. Rev. Letters 17 (1966) 716, D. J. Mellema, C.Y. Chien, W. E. Slater, D.H. Stork, H.K. Ticho, Nucl. Phys. B27 (1971) 437. [11] S. Pokorski and H. Satz, Nucl. Phys. BI9 (1970) 113. [12] R. Carlitz, S. Frautsehi and G. Zweig, Phys. Rev. Letters 23 (1969) 1134. [13] K. Kajantie and S. Popageorgian, Nucl. Phys. B22 (1970) 3. [14] J. C. Berlinghieri, M.S. F a r b e r , T. F e r b e l , B . E . Forman, A. C. Melissinos, P. F. Slattery, T. Yamanouchi and H. Yuta, Phys. Letters 27B (1968) 665.